Internal problem ID [8893]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1314.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (x^{2}+1\right ) y^{\prime \prime }-\left (2 \left (n -1\right ) x^{2}+2 n -1\right ) y^{\prime }+\left (v +n \right ) \left (n -1-v \right ) x y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 35
dsolve(x*(x^2+1)*diff(diff(y(x),x),x)-(2*(n-1)*x^2+2*n-1)*diff(y(x),x)+(v+n)*(-v+n-1)*x*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} x^{n} \LegendreP \left (v , n , \sqrt {x^{2}+1}\right )+c_{2} x^{n} \LegendreQ \left (v , n , \sqrt {x^{2}+1}\right ) \]
✓ Solution by Mathematica
Time used: 0.096 (sec). Leaf size: 75
DSolve[(-1 + n - v)*(n + v)*x*y[x] - (-1 + 2*n + 2*(-1 + n)*x^2)*y'[x] + x*(1 + x^2)*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \, _2F_1\left (\frac {1}{2} (-n-v),\frac {1}{2} (-n+v+1);1-n;-x^2\right )+c_2 x^{2 n} \, _2F_1\left (\frac {n-v}{2},\frac {1}{2} (n+v+1);n+1;-x^2\right ) \\ \end{align*}